1. What Is a Determinant?
A determinant is a single numerical value associated with a square matrix. It helps us understand many things about the matrix, such as whether it is invertible, how it transforms space, and whether its rows or columns are linearly independent.
If a matrix is written as \( A \), its determinant is written as \( \det(A) \) or \( |A| \).
2. Determinant of a 1×1 Matrix
For a 1×1 matrix:
A = [a]
The determinant is simply the number itself:
|A| = a
3. Determinant of a 2×2 Matrix
For a 2×2 matrix:
A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}
The determinant is calculated as:
|A| = ad - bc
This is the simplest and most widely used determinant formula.
3.1. Example
Let:
A = \begin{bmatrix} 3 & 4 \\ 2 & 5 \end{bmatrix}
Then:
|A| = (3)(5) - (4)(2) = 15 - 8 = 7
4. Determinant of a 3×3 Matrix
For a 3×3 matrix:
A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}
One easy method is the Rule of Sarrus, which works only for 3×3 matrices.
4.1. Rule of Sarrus (Simple Pattern)
Write the first two columns again to the right, then add products of the diagonals going downward and subtract the upward diagonal products.
|A| = aei + bfg + cdh - ceg - bdi - afh
4.2. Example
Let:
A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}
Using the formula:
|A| = (1)(5)(9) + (2)(6)(7) + (3)(4)(8) - (3)(5)(7) - (2)(4)(9) - (1)(6)(8)
= 45 + 84 + 96 - 105 - 72 - 48
= 0
The determinant is 0, meaning the matrix rows are linearly dependent.
5. General Expansion for Any Size (Laplace Expansion)
The determinant of a matrix of any order can be found by expanding along any row or any column.
If we expand along the first row of an \( n \times n \) matrix:
|A| = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} + \cdots + a_{1n}C_{1n}
where \( C_{ij} \) are cofactors.
5.1. Example (Expanding a 3×3 Determinant)
Let:
A = \begin{bmatrix} 2 & 1 & 3 \\ 0 & 4 & 5 \\ 1 & -2 & 2 \end{bmatrix}
Expanding along the first row gives:
|A| = 2 \begin{vmatrix} 4 & 5 \\ -2 & 2 \end{vmatrix} - 1 \begin{vmatrix} 0 & 5 \\ 1 & 2 \end{vmatrix} + 3 \begin{vmatrix} 0 & 4 \\ 1 & -2 \end{vmatrix}
You would then compute each 2×2 determinant and add/subtract accordingly.